The periodic dilation completeness problem: cyclic vectors in the Hardy space over the infinite‐dimensional polydisk
The periodic dilation completeness problem: cyclic vectors in the Hardy space over the infinite‐dimensional polydisk
复制标题
DOI：
10.1112/jlms.12365
复制
发表时间：
2019-08
期刊：
影响因子：
--
通讯作者：
H. Dan;K. Guo
中科院分区：
文献类型：
--
作者：
H. Dan;K. Guo
经典完备性问题由Beurling提出，并由Wintner独立提出，要求对于哪一个λ ∈L2（0，1），伸缩系统{λ（kx）：k= 1，2，.}在L2（0，1）中是完备的，其中λ通过它在R上的奇2周期函数的扩张来确定。这个困难的问题现在通常被称为周期性扩张完备性问题（PDCP）。利用Beurling的思想和Bohr变换的应用，将PDCP问题转化为无限维多圆盘上哈代空间H∞2中的循环向量的坐标乘算子的等价刻画问题.本文得到了哈代空间H∞2中循环向量的许多新结果.在几乎所有有趣的情况下，我们得到了充分和必要的标准，为特征循环向量，因此，在这些情况下，我们完全解决了PDCP。我们的结果几乎涵盖了所有以前已知的结果在这个问题上。
The classical completeness problem raised by Beurling and independently by Wintner asks for which ψ∈L2(0,1) , the dilation system {ψ(kx):k=1,2,…} is complete in L2(0,1) , where ψ is identified with its extension to an odd 2‐periodic function on R . This difficult problem is nowadays commonly called as the periodic dilation completeness problem (PDCP). By Beurling's idea and an application of the Bohr transform, the PDCP is translated as an equivalent problem of characterizing cyclic vectors in the Hardy space H∞2 over the infinite‐dimensional polydisk for coordinate multiplication operators. In this paper, we obtain lots of new results on cyclic vectors in the Hardy space H∞2 . In almost all interesting cases, we obtain sufficient and necessary criterions for characterizing cyclic vectors, and hence in these cases we completely solve the PDCP. Our results cover almost all previous known results on this subject.